Neurulation and the cortical tractor model for epithelial folding

J. Embryol. exp. Morph. 96, 19-49 (1986) 19

Printed in Great Britain © The Company of Biologists Limited 1986

Neurulation and the cortical tractor model forepithelial folding

ANTONE G. JACOBSON1, GEORGE F. OSTER2,GARRETT M. ODELL3 AND LOUIS Y. CHENG4

1 Center for Developmental Biology, and Department of Zoology,University of Texas, Austin, TX 78712, USA2Departments of Biophysics and Entomology, University of California, Berkeley,CA 94720, USA3Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy,

NY 12181, USA.

^Department of Civil Engineering, University of California, Berkeley, CA 94720,USA

SUMMARY

We present here a new model for epithelial morphogenesis, which we call the 'cortical tractormodel'. This model assumes that the motile activities of epithelial cells are similar to those ofmesenchymal cells, with the added constraint that the cells in an epithelial sheet remain attachedat their apical circumference. In particular, we assert that there is a time-averaged motion ofcortical cytoplasm which flows from the basal and lateral surfaces to the apical region. Thiscortical flow carries with it membrane and adhesive structures that are inserted basally andresorbed apically. Thus the apical seal that characterizes epithelial sheets is a dynamic structure:it is continuously created by the cortical flow which piles up components near where they arerecycled in the apical region. By use of mechanical analyses and computer simulations wedemonstrate that the cortical tractor motion can reproduce a variety of epithelial motions,including columnarization (placode formation), imagination and rolling. It also provides amechanism for driving active cell rearrangements within an epithelial sheet, while maintainingthe integrity of the apical seal. Active repacking of epithelial cells appears to drive a number ofmorphogenetic processes. Neurulation in amphibians provides an example of a process in whichall four of the above morphogenetic movements appear to play a role. Here we reexamine theprocess of neurulation in amphibians in light of the cortical tractor model, and find that itprovides an integrated view of this important morphogenetic process.

INTRODUCTION

In this paper we propose a novel mechanism for epithelial morphogenesis,

which we call the 'cortical tractor' (Jacobson, Odell & Oster, 1985). This model

resolves several paradoxical features of epithelial sheets, including the ability of

epithelial cells to change their positions relative to their neighbours without

Key words: neurulation, modelling, epithelial folding, cortical tractor.

20 A. G. JACOBSON, G. F. OSTER, G. M. ODELL AND L. Y. CHENG

breaking the apical seal, and the ability of epithelial sheets to fold and invaginatein a coordinated fashion over large areas.

During early embryogenesis cell types may be classified into two general groups:mesenchyme and epithelia (Hay, 1968). Epithelial cells are organized into cohes-ive sheets that cover surfaces and line cavities. Each epithelial cell has a definitepolarity, with the apical surface facing the external medium or the internal cavity.Just beneath the apical surface a circumferential complex of junctions bonds theepithelial cells tightly to one another and seals the embryo from the outsideenvironment, or from the contents of the cavities. The basal surface usually faces abasal lamina and other extracellular matrix material. When epithelial cells changeshape, the sheet in which they are embedded may roll, fold, invaginate orotherwise deform while retaining its integrity as a connected tissue.

Mesenchymal cells are loose, migratory cells that frequently move aboutbeneath and between epithelial layers. The polarity of mesenchymal cells reflectstheir current direction of movement, which changes as the cells move about. Theirmigratory motions often appear relatively independent of one another, but theyfrequently aggregate to form patterned clusters of cells.

Embryonic morphogenesis results from the coordinated behaviour of epithelialsheets and groups of mesenchymal cells. However, the identity of these two celltypes is not immutable; frequently during early embryogenesis cells convertbetween epithelial and mesenchymal forms (Hay, 1968). For example, epithelialcells in the epiblast of bird embryos converge to the primitive streak, pass throughand detach from the epithelium to wander beneath as mesenchyme cells. Some ofthese form orderly clusters of cells (somitomeres) that eventually condense intoepithelial somites (Meier, 1984). Later, cells in regions of the epithelial somitesemerge again as mesenchyme, which then migrate and cluster to form bones, skin,or muscle masses.

Clearly, the same cytoskeletal machinery used by mesenchyme cells to moveabout is present in epithelial cells, albeit organized differently. We shall proposethat epithelial cells do indeed move about in much the same fashion as mes-enchymal cells, changing neighbours and deforming the epithelial sheet whilemaintaining, indeed constructing, the apical seal. The mechanism by whichmovement and morphogenesis are accomplished we call the cortical tractor; wepropose it as the driving force underlying epithelial morphogenesis, includingepiboly, gastrulation, neurulation, placode invagination and evagination. Here weshall apply the cortical tractor concept to explain certain paradoxical features ofamphibian neurulation.

GENERAL FEATURES OF CELL MOTIONS

Motility, the capacity to move and change shape, is a fundamental property ofcells, and all cells in the embryo are potentially motile. Mesenchymal cellsregularly move about, and epithelial cells, when disaggregated, also move aboutmuch like mesenchymal cells (Holtfreter, 1946; Middleton, 1973). Cells at and

Cortical tractor model for epithelial folding 21

near the free edges of an epithelial sheet are motile, and their activities mayexpand the sheet (Vaughn & Trinkaus, 1966). Examples of this are the expansionof the chick blastoderm (New, 1959; Downie & Pegrum, 1971), and wound healing(Radice, 1980a, b). Epithelial cells may also move about amongst one anothertangentially within an intact epithelial sheet. These movements may be organizedso as to produce specific deformations of the sheet. This is a particularly para-doxical phenomenon, and one to which the cortical tractor model proposes asolution.

While there are differences in the details of how different cell types move(Trinkaus, 1984a), there are fundamental similarities. Oster (1984) has proposed amodel for lamellipodial motion that ties together many properties of moving cells.In this model, the cell's movement is driven by the actomyosin cortical gel, whichundergoes a cycle of events involving solation and osmotic expansion followed byregellation and contraction. As these events proceed, there is a fountainoid flow ofcytoplasm from the cell interior to the region where the cell's anterior is pro-truding, and a flow of the cortex away from that region. Elsewhere in the cell thecortical material is again recycled into the cell's interior. Osmotic expansion of thepartially solated gel at the lamellipod tip, together with hydrostatic pressurecreated by contraction of the cortical gel elsewhere, extends the tip of thelamellipod. Cell and substrate adhesion molecules are carried by the fountainoidof cytoplasm to the cell surface at the tip of the lamellipod and inserted into themembrane. These molecules adhere to adjacent cells, or to the substrate, and areattached through the membrane to the cytoskeleton of the cortex where theyanchor the newly extended lamellipod tip. Thus when the cortical actin regels andcommences active contraction, the cell inches forward. In this model, the tip of thelammellipod is a special site for ionic leaks that lead to an increase in free calciumin the tip cortex; it is this elevated cortical calcium concentration that initiates andmaintains the solation-contraction cycle just described.

While the details of this particular model may not apply to all cells (or even toany cells), what is crucial to our discussion are the following features of cellmotion: (a) the overall flow pattern of cytoplasm, and (b) the insertion of adhesionmolecules in the leading edge. In the following section we will extract theseproperties and formalize them as a set of postulates that embody the corticaltractor model.

THE CORTICAL TRACTOR MODEL

The cortex of a moving cell flows in a fountainoid pattern

The cortical tractor model will apply to all motile cells, whether crawling asindividuals or in groups; in particular, it will apply to cells joined into an epithelialsheet. The first postulate of the cortical tractor model is:

Cell motion is characterized by a 'fountainoid'flow pattern of cytoplasm from theleading to the trailing edge of the crawling cell, as shown in Fig. 1. We shall callthis flow pattern the 'cortical tractor'.


Several authors have suggested, and given evidence for, this type of corticalmotion (e.g. Allen, 1961; Abercrombie, Heaysman & Pegrum, 1970; Harris, 1973;Odell & Frisch, 1975; Abercrombie, 1980; Dembo & Harris, 1981; Bretscher,1984). Abercrombie et al. (1970), Harris & Dunn (1972) and Dembo & Harris(1981), among others, did marking experiments that demonstrated that the cor-tical flow is reflected on the surface of the cell, indicating an intimate connectionbetween the flowing cortex and the plasma membrane. Bretscher (1984) focuseson the flow of the membrane and its return to the interior via endocytotic vesicles.He suggests that the motion of the cell is the result of membrane flow; however,since the plasma membrane is fluid, it does not have the mechanical propertiesnecessary for cell movement (see, for example, the discussion by Ambrose afterthe paper by Petris & Raff, 1973).

In our model, the essential feature is the flow pattern of the cortical gel. Themembrane may or may not follow the cortical flow; its essential role in the settingof the model is to regulate the communication between the cell and its ionicenvironment. Also, a particular cell may generate more than one tractoring locus;indeed, any motile region of a cell's cortex will undergo local tractoring in apattern analogous to that shown in Fig. 1.

Adhesion structures are cycled with the cortical flow

In order for the cortical tractor to drive cell movement there must be adhesionstructures anchoring the cell to its substrate or to neighbouring cells. The corticaltractor model requires that these adhesive molecules are cycled along with thecortex (Campbell & Campbell, 1971). Therefore, our second postulate is:

Adhesion and functional molecules are inserted at the site of the cortical 'source'(the cell's leading surface). They flow posteriorly with the cortex and are resorbedat the cortical 'sink', unless stabilized by bonding with the substratum, or withanother cell

Sink

Time-averagedcortical flow

Active end(Source)

Fig. 1. Schematic of the cortical tractor mechanism. The actomyosin cortex has atime-averaged flow as indicated by the arrows, and the cell-to-cell adhesion structuresare dragged along by the flow of the cortex.


A moving cell attaches itself to its substratum by attachment sites that arecontinuously being inserted at the leading edge. These sites are dragged towardsthe cell's posterior where they are resorbed. Cell-cell and cell-substrate adhesionsites pass through the membrane, attaching the cell's cortex to the point ofadhesion outside the cell. Being a lipid fluid, the membrane can flow around theattachment sites, or alternatively the sites may be dragged through the fluidmembrane. Junctional structures inserted at the leading edge of the cell flow withthe cortex to the trailing edge, where they may become a transient part of specificjunctions, such as desmosomes (Campbell & Campbell, 1971).

Ionic stimuli may activate the cortical tractor

The cortical tractor may be activated by ionic stimuli. Experiments withionophores and channel-plugging molecules show that cell motility can be stimu-lated by ionic leaks, especially of calcium, and inhibited by blocking ionic channels(Zigmond, 1978; Snyderman & Goetzl, 1981; Cooper & Schliwa, 1985). Anythingthat initiates ionic leaks at a cell surface (or blocks such leaks) could start (or stop)cell movement in that direction. The direction of movement (i.e. the orientation ofthe cortical source) is an important part of the cortical tractor model. There isample evidence for cellular and molecular mechanisms that regulate ionic con-ditions, although it is not yet clear how they are mediated. The proximal signal foractivating the surface of a cell could be nonionic chemical messengers such aschemotactic agents that bind specific receptors on the cell surface and initiate ionicevents that trigger motile activity. A variety of mechanisms and components mayregulate calcium and other ions that affect motility (Hitchcock, 1977; Oster, 1984).We need not be specific about these mechanisms here, for whatever the detailsmay be, the only essential aspect for the cortical tractor model is that motility canbe triggered by the local chemical environment.

Epithelial cells remain attached to one another at their apical ends

The final ingredient of the cortical tractor model of epithelial morphogenesisrequires that epithelial cells have some mechanism of maintaining their integrity asa cell sheet. Thus our third postulate is:

Cortical tractoring occurs in epithelial cells much as it does in mesenchymal cells,except that epithelial cells remain firmly attached to one another at their apicalboundaries.

The cortical tractor suggests a mechanism for this attachment: if the recyclingrate for adhesion structures is slower than their insertion at the cortical source (i.e.the basal surface), then they will 'pile up' at the apical end, and prevent the cellsfrom detaching.

Epithelial sheets are dynamic structures

The cortical tractor model views an epithelial sheet not as a static cobblestonepaving of cells, but as a dynamic structure that is in constant motion. As shown in


Neural plate surface

Apical Junctionalcomplex

Fig. 2. Schematic of a cross section of an epithelial sheet. The time-averaged flow ofthe cortex in each cell comprises the.cortical tractor motion (arrows). This cortical flowinserts adhesive structures at the basal source, and reabsorbs them in the apical region.These junctional molecules pile up at the apical end and form the apical seal, astructure that is constantly being renewed. As long as the time-averaged speed of thecortical tractor in adjacent cells remains approximately equal, no shear forces will begenerated, and no deformation of the epithelial sheet will take place.

Fig. 2, each cell is constantly tractoring its cortex from basal (or basolateral) toapical, and the apical seal is continuously renewed by the insertion and recycling ofadhesive structures (e.g. Larsen & Risinger, 1985). As we mentioned earlier, thereis evidence that suggests that more than one tractor site can coexist on a given cell.For example, different virus particles will insert specifically at basal, lateral orapical surfaces of an epithelial cell, and exocrine cells secrete through their apicalsurfaces; therefore, there can be a basal, apical and, or, lateral tractor operating atvarious times in a cell. Therefore, the time-averaged flow of cortical cytoplasm islikely to be more complicated than a simple basal-to-apical motion; however,many of the phenomena we shall discuss can be understood in terms of thesimplified cortical flow shown in Fig. 2.

Differences in cortical tractor velocity will deform an epithelial sheet

The apical surface of an epithelial cell does not become an active, leading edgeexcept in unusual circumstances, and then the cell is likely to crawl out of theepithelium (cf. Holtfreter, 1947). However, any other free surface of the cell,either basal or lateral, or both, may be active. Each cell in Fig. 2 is attempting tocrawl downward on its neighbours by adhesion to their surfaces and contraction ofits cortex. However, if adjacent cells are equally active in their crawling motions, no nettraction can be developed between them, and no net motion results. If the intensity ofcrawling (i.e. the speed of the cortical tractor) is different between adjacent cells,shear forces will be generated and the faster cell will commence to crawlbasal ward, out of the sheet. The apical junctional complexes restrain each cellfrom detaching from the epithelial sheet, and so the net effect of differentialcrawling is that cells crawling most rapidly elongate in the basal direction. Thusimposing a gradient of tractor velocity will convert cuboidal epithelial cells into acolumnar morphology.

Of course, a verbal statement of such a course of events cannot be convincing,and so we have constructed a mechanical model to demonstrate how an epitheliumconsisting of cuboidal cells can generate placode regions of columnar cells by


differences in the speed of the cortical tractor (Cheng, Murray, Odell & Oster,1986; a sketch of this model is given in Appendix A).

Epithelial morphogenesis is driven by cell shape change

An epithelial layer as a whole can deform by spreading, retracting, folding,bending, rolling into a tube, evaginating or invaginating. As such deformationsproceed, the constituent cells of the epithelium change shape. Indeed, it is fair toassume that the global deformations of the sheet are driven by the shape changesof the individual cells. In order to change its shape, an epithelial cell must generateinternal forces to work against the tractions of its neighbouring cells. Bundles ofmicrofilaments are often found encircling the apical surface of cells in a deformingepithelium, and the constriction of these filament bundles is thought to contractthe apical surfaces to create wedge-shaped cells which would roll the epithelialsheet (Baker & Schroeder, 1967; Burnside, 1971; Odell, Oster, Alberch &Burnside, 1981).

The cortical tractor mechanism dramatically augments the apical filamentbundle hypothesis by providing an additional means by which epithelial cells canchange shape. Indeed, the cytoplasmic flow pattern associated with the corticaltractor mechanism suggests that the microfilaments may have accumulated in theapices by the tractor motion. Moreover, the cortical tractor mechanism canreproduce all of the epithelial foldings associated with the apical contractionmodel (Odell et al. 1981) as well as many deformations that apical contractionalone cannot produce. (For example, the cuboidal-to-columnar transitions charac-terizing placode formation which precede most foldings cannot easily be generatedby apical contractions.)

Epithelial cells can change neighbours without breaking the apical seal

One of the most paradoxical properties of cells in an epithelium is their ability tomove within the plane of the sheet, thereby changing their neighbour relation-ships, without violating the integrity of the apical seal. For example, neighbourchanges are frequently seen in apical views of the midline of the neural plate(Jacobson & Gordon, 1976), and in the surface epithelium near the blastoporeduring amphibian gastrulation (Keller, 1978). Such cell rearrangements have alsobeen seen in the epithelia of insect imaginal discs (Fristrom, 1982) and in theenveloping layer of Fundulus (Trinkaus, 19846).

The general situation is shown in Fig. 3A: cells b and d do not contact oneanother initially, but at a later time they come to share a common boundary. Anykind of epithelial rearrangement can be built up from such binary neighbourChanges. The cortical tractor mechanism suggests that the answer to such apicalreorganizations lies below the surface, as shown in Fig. 3B. Cell b becomes activeon its lateral face and puts out a protrusion toward cell d. As the lamella is carriedapicalward by the cortical flow, it eventually appears from above as a cytoplasmicextension extruding itself between cells a and c. That is, an apical view would give


the impression that b had crawled between a and c, and so must have broken theirapical bonds. However, this is an illusion, for no bonds need be broken. Ratherthe flow of new adhesive and junctional structures up from below replaces thejunctions being recycled to the interior of the cells. Thus the apical seal is neverbroken, as the cells actively interdigitate between one another.

Cell repacking may drive other morphogenetic processes

There is a growing body of observations that support the conclusion thatepithelial sheets can alter their global geometry by an active repacking of theirconstituent cells. This may underlie such diverse phenomena as the evagination ofimaginal discs in Drosophila (Fristrom, 1982), and late gastrulation in sea urchins(Ettensohn, 1985; Harding & Cheng, 1986). Indeed, epiboly (the spreading andflattening of an epithelial sheet) generally involves cell rearrangements, althoughin some cases it appears as if they may be passively driven by forces developed atthe edge of the sheet (e.g. epiboly in Fundulus, Trinkaus, 1984b). The corticaltractor mechanism provides a means for cells in an epithelial layer to repackactively, and in so doing drive large-scale changes in the geometry of the sheet.How these active repackings are coordinated so as to arrive at the desired endconfiguration may vary from tissue to tissue. In the next section we suggest howthis coordination can be achieved during neurulation.

(A) Apical views

(B) Lateral views

d a b d a b

Fig. 3. (A) An apical view of four epithelial cells undergoing binary neighbourexchange. Cells a and c lose their shared apical boundary, and cells b and d cometogether to form a neighbour pair. (B) A lateral view of how the cortical tractormechanism effects neighbour exchanges. Cell b interdigitates between cells a and c byextending lateral lamellae (L) which are swept upwards by the cortical motion. L carrieswith it new adhesive structures which pile up at the bottom of the apical junction,replacing ingested junctional structures. When L reaches the apical surface it appearsfrom above as an extension of cell b which is interdigitating between cells a and c.


Summary of the cortical tractor model

In summary, the cortical tractor model for epithelial motility is composed of thefollowing sequence of events.

(1) A change in the local chemical environment initiates motile activity in thebasal and, or, lateral surfaces of epithelial cells, and lamellae commence toextrude from the active surfaces.

(2) The flow of cytoplasm from the interior of the cell brings adhesive andjunctional structures to the cell surface at the source, where they insert through themembrane and adhere to the substratum or to adjacent cells.

(3) Cytoplasm flows into the protrusions at the active face of the motile cell,gels and contracts. This produces an average movement of cortical cytogel thatflows in a fountainoid pattern (cf. Fig. 1) from basal/lateral to apical.

(4) Adhesive structures are swept by the cortical flow apicalward where theypile up and form the apical seal, eventually being resorbed into the cell interior.

(5) Adjacent cells tractoring at the same velocity cannot develop any shearforces. Therefore an epithelial sheet, whose cells are all tractoring with the sameintensity, will appear as a static structure. However, if a velocity gradientdevelops, shear forces are generated that deform the epithelial sheet as each cellattempts to crawl out of the layer on its neighbours.

In the next section we apply these ideas to the process of neurulation, and showhow boundaries between domains of different cell types act as organizing centresto coordinate the shear forces developed by the cells of the neural plate.

NEURULATION

In the embryos of amniotes and many amphibia the neural ectoderm is a simple,or single-layered, neuroepithelium. It first appears as a region of cells thatcolumnarize, becoming taller than the rest of the ectoderm (Fig. 4).

In mammals (Jacobson & Tarn, 1982) and birds (Schoenwolf, 1985), the neuralplate grows during neurulation; however, amphibia are simpler to study becauseneurulation proceeds without growth (Jacobson, 1978). Some amphibia have verylarge cells with variegated egg pigmentation that makes direct observation ofindividual cells possible. Thus we shall use neurulation in the California newtTaricha torosa as a prototype of cortical-tractor-driven morphogenesis (Jacobson,1981). Developmental stages for this species were described by Twitty & Boden-stein, and are illustrated in Rugh (1948).

During neurulation a number of morphological changes occur in the neuralplate:

(1) The cells of the plate change shape, first changing from cuboidal to colum-nar, and finally becoming wedge-shaped.

(2) The width of the neural plate decreases, and its total surface area shrinks.(3) The midline of the plate elongates, and cells of the notoplate undergo

systematic neighbour changes.(4) The neural folds form at the plate edges, and subsequently elongate.

STAGE 15

no

nf

Fig. 4. The neural plate of a newt embryo, and the tissues that underlie it, areshown in cross section at stages through the period of neural fold formation and rollingof the plate into a tube. These plastic sections (2/jm thick) are from the region wherethe spinal cord joins the brain. All are at the same magnification. (A) Stage 15;(B) stage 17; (C) stage 19. ep, epidermis; me, mesoderm; nc, notochord; nf, neuralfold; no, notoplate; np, neural plate. Bar, 0-1 mm.


(5) The plate rolls into a tube.Each of these changes contributes to the shaping of the plate and the formation

of the neural tube. In this section we shall argue that the cortical tractor is acontributing force to all of these events.

Cell shape changes accompany neurulation

The cells of the neural plate continue to get taller throughout the course ofneurulation, but cells in different regions of the plate seem programmed to arriveat characteristically different heights by the end of the process (Jacobson &Gordon, 1976). The specific cell shape changes can be followed only in a mappedsystem because cells move long distances during neurulation. Studies that purportto describe changes in the shapes of the neural plate cells during neurulation byexamining sections (e.g. Baker & Schroeder, 1967) are really describing the shapesof different groups of plate cells at different times. This is because there is acontinuous flow of cells through the site of sectioning during neurulation. Burnside& Jacobson (1968) mapped the changes of cell position in the neural plate ofTaricha torosa by analysing time-lapse films of neurulation. Burnside (1971, 1973)has correlated apical constriction of the cells to circumferentially organized purse-strings of microfilaments near the apical surface, and elongation to microtubulesoriented along the long axes of the cells. She gives evidence that the apicalmicrofilament bundles may be contracting to reduce the surface area, and thatmicrotubules may play a role in cell elongation, perhaps by directing cytoplasmicflow toward the basal ends of the cells (Burnside, 1971).

Neural plate cells change neighbours during neurulation

Changes of cell shape, driven by apical contraction and/or the cortical tractor,play a central role in neurulation. However, certain cell groups in the neural plateappear to be more active in changing neighbours than in changing shape, and theconsequences of their movements are essential for shaping the plate. Jacobson &Gordon (1976) found that the neural plate consists of two populations of cells. Onegroup of cells overlies the notochord, and the other comprises the rest of theneural plate. We will refer to these populations as notoplate and neural plate,respectively.

Convergence and extension of notoplate is crucial in shaping the plate

In the early gastrula, the prospective notoplate is a crescent of tissue betweenthe prospective neural plate and the prospective notochord. The notochord anlagesits on the dorsal lip of the forming blastopore (Fig. 5).

The notochord involutes around the blastopore lip, then converges toward andextends along the midline. The notoplate does not involute, but remains on thesurface as part of the neural plate. It undergoes the same convergence-elongationmovements as the notochord that underlies it.

The notoplate begins as a crescent of cells near the blastopore, then convergestoward and elongates along the midline, a process which distorts the geometry of


the plate (Fig. 5). In an experimental study and computer simulation, Jacobson &Gordon (1976) found that the early shaping of the neural plate into a keyholeshape is largely due to the elongation of the notoplate, but also requires apicalconstriction of the plate cells in order for the plate shape to come out normal.

The elongation of the midline is even more dramatic during closure of thekeyhole-shaped plate into a tube. During that period, elongation of the plate is tentimes as fast as before or after (Jacobson & Gordon, 1976). Jacobson (1978)proposed that midline elongation of the plate could shape it into a tube by thesame mechanism that rolls a rubber sheet when it is stretched along a line (so-called 'Poisson buckling').

Notoplate cells and neural plate cells are two separate populations

Besides the differences in behaviour described between notoplate and neuralplate cells, there are other observations that suggest these are quite differentpopulations of cells. For example, the notoplate cells look different from the cellsof the rest of the neural plate (Fig. 6).

Keller and coworkers (Keller, Danilchik, Gimlich & Shih, 1985) explantedpieces of the dorsal lip of the blastopore of early gastrulae of the anuran Xenopuslaevis. In culture, the explanted dorsal lip displayed four distinct regions, eachwith different behaviours. Prospective bottle-cells and head mesoderm from thelip of the forming blastopore continued their normal development in culture. Cellsin the region of the prospective notochord underwent convergence and extensionalong a line, just as they would have in the embryo. A third group of cells formsthe prospective notoplate, and these cells too have the intrinsic capacity to executethe convergence and elongation movements. The final region in the explant is theprospective neural plate and, in the absence of underlying inductors, these cellsform a vesicle of indifferent ectoderm. It should be noted that the intrinsicbehaviour of the notoplate cells does not require induction by underlying tissues.

The notoplate behaves like the notochord, except it does not involute. It mayhave been induced earlier along with the notochord (Gerhart, 1980). In theexperiments of Spemann & Mangold (1924) that won the Nobel Prize, a piece ofthe dorsal lip of the blastopore of an early gastrula of Triton cristatus wastransplanted into the ventral ectoderm of a Triton taeniatus host embryo. A secondneural plate was induced in the ventral ectoderm of the host embryo by theimplanted dorsal lip material. Differences in pigmentation between donor andhost species allowed the tissues of the two species to be distinguished from oneanother. At least in some cases, the induced neural plate formed from host tissue,except for the midline region (i.e. the notoplate) that came from the graftedimplant (Spemann, 1938, his fig. 78, p. 144). The donor notoplate cells must haveinsinuated themselves down the midline of the induced host neural plate. InSpemann's illustrations of sections through the spinal cord of later stages, whenthe secondary nervous system had formed into a tube, the donor notoplate tissueappears as the floor plate of the spinal cord (Spemann, 1938, his fig. 80, p. 146).The notoplate, in fact, maps congruent with the floor plate of the nervous system.


Neural plate

Notoplate

Notochord

Blastopore lip

Notoplate

Neuralfolds

Early gastrula Late gastrula Mid-neurula Late neurula

Fig. 5. The parts of the embryo referred to in the text are labelled on these drawings ofa rear view of an early gastrula, and dorsal views of late gastrula to late neurula of anewt embryo. The notochord shown above the dorsal lip of the blastopore in the earlygastrula has involuted around the blastopore lip, and in subsequent stages lies beneaththe notoplate. Figures are approximately to scale. An early gastrula is about 2-5 mm indiameter.

The fate of the floor plate is to become a raphe (i.e. a seam) between the basalplates in the spinal cord and brain. This is precisely the ultimate consequence ofthe cell behaviour we propose for the notoplate.

The boundary between notoplate and neural plate is an important organizing regionfor neurulation

Jacobson (1978) found an inverse correlation through time between the lengthand width of the neural plate of Taricha torosa. The implication was that cells wereinterdigitating themselves so that the plate length increased at the expense of itswidth. Cell counts of cross sections of the plate reinforced this interpretation. Thenumber of cells in cross sections from a given level of the plate diminished at ratesthat were inverse to the rate at which the plate elongated.

The boundary between notoplate and neural plate gets progressively longerduring neurulation (Fig. 5). The change in the length of this boundary, asmeasured from time-lapse films of neurulation, was one of the quantitative inputsto the computer simulations of Jacobson & Gordon (1976). They suggested thatthe boundary elongated as cells from the interior of the notoplate region re-positioned themselves along the boundary.

Adhesive differences between notoplate and neural plate cells are important

Jacobson (1981, 1985) has suggested one means by which this repositioningcould come about. He proposed that the adhesive differences between thenotoplate cells and the neural plate cells are such that the two cell types shouldmix, but that the neural plate was more 'solid' than the 'fluid' notoplate area. Thusactual mixing was difficult, and instead the adhesions between notoplate andneural plate cells were maximized by elongating the boundary. For this to happenit would be necessary that the cells move about somehow so that new contacts

32 A. G. JACOBSON, G. F . O S T E R , G. M. O D E L L AND L. Y. C H E N G

Fig. 6. This scanning electron micrograph shows a view of the bottom and broken faceof the neural plate of a Xenopus laevis embryo. A piece of the notochord has brokenaway to reveal the bottoms of the notoplate cells and their relationships to matrix andto the notochord. The notoplate appears distinctly different from the rest of the neuralplate in both bottom and edge views, nc, notochord; no, notoplate; np, neural plate.Bar, 10|Um.

would be made between notoplate and neural plate cells. The cortical tractormechanism provides a means for cells to move about in an epithelium, and inparticular a means for active intercalation along the notoplate-neural plateboundary. Coupled with adhesive differences between groups of cells, this endowsthe boundary between notoplate and neural plate with a central organizing roleduring neurulation.

The cortical tractor can elongate the notoplate-neural plate boundary

Notoplate cells attach intimately at their basal surfaces to the underlyingnotochord, while more lateral neural plate cells hang free in the extracellular spacewhere they are probably in contact with extracellular matrix (Fig. 4). Apical


surfaces are normally inactive in epithelia; if contact with the notochord inacti-vates the basal surfaces of notoplate cells, then notoplate cells can only be activeon their lateral faces. Thus one would expect that their cortical tractors would runonly laterally. We have examined tangential sections through the notoplate andneural plate and found numerous lateral lamellipodia (Fig. 7), some extendingthree to four cell ranks away from the cell body. We suggest that theselamellipodia are in fact the organelles which enable the cells to interdigitatebetween their neighbours by flowing apicalward, as we have described.

The basolateral activity of notoplate cells cannot be random

If the direction of tractoring from the lateral faces were random, then the cells ofthe notoplate should execute a random walk, and there would be no net distortionof the notoplate in any direction. There are at least two ways that the boundarybetween the notoplate and the rest of the neural plate could organize thetractoring of the notoplate cells. If notoplate cells that contact neural plate cellsadhere to them more strongly than they adhere to each other (differentialadhesion), or if contact with a neural plate cell surface inhibits the activity of thatface of the notoplate cell (contact inhibition), then tractoring in the notoplate cellwill be restricted to those faces not abutting the neural plate. Cells that have justcollided often form a new active surface on the face opposite the collision(Trinkaus, 1984a, p. 354; Oster, 1984). The result is that cells stuck at the boundarywould tractor additional cells toward the boundary, causing them to interdigitatealong the boundary and elongate it, as shown in Fig. 8.

Once a notoplate, or neural plate, cell contacts the boundary, it will remain onthe boundary thereafter. Some preliminary analyses of time-lapse films of neuru-lation in Taricha torosa support this prediction (Fig. 9). As directed cell inter-digitation elongates the notoplate boundary, the adjacent tissues on both sides willbe stressed, thus facilitating the interdigitation of further ranks of cells along theaxis of the embryo.

We have measured a four-fold increase in the length of the notoplate duringneurulation. Interdigitation of just four ranks of cells along the embryonic axisin the notoplate is sufficient to account for this elongation. However, duringneurulation there are many more than four ranks of cells in the notoplate, andshortly after neurulation is complete, the descendent of the notoplate, the floorplate, is still four ranks wide at stage 22. The neural tube elongates considerably,and by stage 26, the floor plate is but one cell wide, and that cell is stretchedperpendicular to the embryonic axis.

Between stages 13 and 15 the neural plate changes from a disc to a keyholeshape, but the boundary between the edge of the plate and the epidermis remainsa constant length (Jacobson & Gordon, 1976). However, at stage 15 this boundarycommences to rise and form the neural fold, and the plate begins to roll into atube. Between stages 15 and 20, as the plate is rolling into a tube, the neural foldselongate by about 30 %. Jacobson & Tarn (1982) found that while the brain plateof the mouse embryo rolls into a tube, the neural folds elongate but the midline

34 A. G. JACOBSON, G. F. O S T E R , G. M. O D E L L AND L. Y. C H E N G

Fig. 7. Tangential, or frontal, section through the neural plate of a stage-17 newtembryo, about halfway down through the thickness of the notoplate. Numerouslamellipodia (arrows) can be seen extending laterally several cell diameters from thebodies of the notoplate cells. The neural plate-notoplate boundary runs across thetop of the figure. The plate was removed from underlying tissues (which opened thespaces between the cells somewhat), then fixed immediately in aldehydes, embeddedin plastic, and sectioned at 2jum thickness. Bar, 0-1 mm.

Neural plate fBoundary

Fig. 8. Schema of how the cortical tractor drives boundary elongation (apical view).The unshaded cells abut the boundary between the notoplate and the neural plate,where their lateral surfaces are deactivated. The other lateral faces continue to tractoras shown, drawing the shaded cell onto the boundary, where it too adheres.


Fig. 9. Cell trajectories have been traced on time-lapse films of early neurulationin the newt embryo. At stage 13 (late gastrula, dashed lines) a cell with distinctivepigmentation was located (labelled Omin). This cell was followed frame-by-frame,and drawn in at new locations at the intervals indicated. In the figure at the left, the cellwas initially within the notoplate area, and had arrived at the notoplate-neural plateboundary by 702 min. (This boundary is drawn in at intervals; it is visible in the film.)Once the cell is on the boundary, it stays there. The solid lines outline the embryo andnotoplate at stage 15, when the analysis was terminated. In the figure at the right, a cellthat began in the neural plate is similarly traced. This cell arrived at the boundary at2002 min, and remained there.

does not. The mouse brain plate, which is very wide, also appears to begin rollingnear the neural folds. It is evident that the neural fold boundary could also be anorganizing line for elongating the neural plate.

The cortical tractor can roll the neural tube while elongating the plate boundary

The cortical tractor model can account for a number of observed cell behavioursat the neural fold boundary. First, we assume that the epidermis is not very activeon the basal surface (nor apically, as usual for ectodermal layers). Thus the cells ofthe epidermis can move only tangentially amongst one another. Second, weassume that the adjacent neural plate cells are basally and laterally active (perhapsbecause of a different basal environment). As the plate cells tractor, the faces thatabut the epidermal cells will adhere to them (or be contact inhibited by them).Since there is a difference in tractor speed between the plate cells and theepidermal cells, the plate cells will crawl beneath the epidermal cells. Thiscrawling will have two consequences.

First, since the apical surfaces of the plate cells are anchored in the epithelialsheet, the cells will be drawn out into long, tapered cells, as shown in Fig. 4(analogous to bottle-cells in the gastrula). The overlying epidermal cells will bedrawn toward the centreline, and a rolling moment will be produced which forcesup the plate edge to form the neural fold. (By rolling moment we mean a torquegenerated by a difference in forces between the apical and basal surfaces.) Thenext rank of plate cells, crawling on the first rank, will commence to elongate andgenerate a further rolling moment, and so on. This sequence of events isdiagrammed in Fig. 10.


Neural plateEpidermis

V

Fig. 10. The diagrams at the left illustrate how we interpret events at the epidermis-neural plate boundary. The drawings on the right are tracings of cells from crosssections of newt neurulae at stages 15, 16, and 17. Because of a difference in tractorvelocities, neural plate cells tractor onto the bottoms of the epidermal cells, pullingthem into a fold. This stretches the cells until the apical surfaces are attenuated almostto points, or even pull loose from the surface. Neural plate cells also interdigitatealong the boundary (interdigitation is not shown in this cross-sectional view) thuslengthening the neural folds. The combination of basal crawling and apical constrictiongenerates a rolling moment that lifts the folds up from the plane of the plate, and rollstoward the midline.

Second, since many ranks of plate cells are crowding onto the boundary region,the boundary will elongate by the same mechanism as the notoplate; that is,epidermal and plate boundary cells contacting one another will have motileactivity suppressed at their contacting faces, leaving only those faces away fromthe boundary active. This effect will spread medially as shown in Fig. 4. Thus dif-ferences in tractor intensity between plate and epidermal cells can both elongatethe plate boundary, and roll the plate into a tube.

Note that the action of the cortical tractor elongates the cells, and constricts theapical boundary (analogous to the tapering of the tail end of a motile mesenchymalcell which remains attached to the substratum). This has the same mechanicaleffect as apical contraction via circumferential filament bundles, which may also beconstricting at this time. The combined effect of apical constriction (via tractoringand, or, filament bundles) and basal crawling is to produce a strong rollingmoment.

As the plate cells crawl underneath the epidermis, and their apical surfacesconstrict, stress is concentrated in the long, thin necks of these bottle-like cells.That is, the traction forces generated by the basal ends will be concentrated in asmaller and smaller apical cross section. Indeed, if the apical ends of the plate cellsbecome sufficiently attenuated, they may accumulate sufficient stress to actually


tear loose from the apical junction structures to form a detached cell population.This may be how neural crest cells separate from the epithelium.

SIMULATION STUDIES OF NEURAL PLATE FOLDING

Neurulation involves the coordination of a number of morphogenetic processes;the three principal cell movements that drive the processes are:

(1) Columnarization of cuboidal cells (i.e. increase in the cell height/crosssectional area ratio) to form the initial neural plate.

(2) Active repacking (i.e. neighbour exchange), especially of notoplate andmarginal cells, which elongate the plate centreline and borders.

(3) Generation of bending moments which roll the plate into a tube.It is not easy to assess the relative importance of the various cellular events listed

above. Therefore, we have resorted to computer simulation studies to try anddeduce how the various processes contribute and fit together. Here we reportsome preliminary results based on finite element models developed by Cheng(1986) and Cheng et al. (1986), which are described briefly in Appendices A and B.

Columnarization

Odell et al. (1981) simulated the neural plate by a finite element model thatinvolved only bending moments generated by apical contraction. They found that,regardless of where the contraction was initiated (e.g. at the centre, or at the plateedges) the first event was the columnarization of the active cells and the flatteningof the active cell population into a plate. Cheng (1986) constructed a model forepithelial deformation driven by the cortical tractor mechanism; a brief descrip-tion is given in Appendix A. This model demonstrates that cell tractoring can alsoproduce columnarization of the plate. Moreover, the model shows how placodes(regions of columnarized cells) arise wherever there is a nonuniformity in tractorvelocity. By our third postulate, tractor velocity will vary where there are gradientsin ionic environments. This suggests that columnarization may correspond toregions where chemical gradients are sustained for some time, perhaps due toaltered cell-cell communication via gap junctions, or to sources and sinks ofchemical 'morphogens'. In the latter case, the cortical tractor 'reads out' itschemical environment in a counterintuitive fashion (cf. Appendix A).

On the basis of the models, therefore, it is only possible to assert that eithermechanism, or both mechanisms (tractoring and apical constriction) acting simul-taneously and uniformly over the presumptive plate, can produce the initiallyflattened neural plate. Nor must one conclude that only these two mechanisms areinvolved in columnarization. For example, Burnside (1971) and others havedocumented the appearance of vertical arrays of microtubules in columnarizedcells, which may play a role in stabilizing the columnar state by 'freezing in' thenew cytoskeletal configuration.


Rolling

The simulations of Odell et al. (1981) were essentially two-dimensional (i.e.plane stress), and neglected hoop stresses. The finite element model of Cheng liftsthese restrictions, but the results, shown in Fig. 11, remain essentially unchanged:bending moments generated either by apical constriction or by tractoring can rollthe neural plate into a tube.

Fig. 11 shows the results of applying a uniform rolling moment across theneural plate, due to basal tractoring and/or apical contraction. The sequence ofdeformed configurations for increasing magnitudes of the applied moment are

Fig. 11. Computer simulations showing how bending moments generated by the basalcrawling of plate cells roll the neural plate model into a tube; see text for discussion.

/1/ I0

11 ,

Jm

1

11

/

Fig. 12. A pair of forces applied to a plate as shown will produce a transverse bucklingforce (Toisson buckling'), as postulated by Jacobson (1981). In this numerical simu-lation, two forces are applied along a line. If the plate is thick enough (left) it willdeform without buckling out of the plane. A thinner plate, however, will buckle up outof its original plane (right). The two views are from the edge (bottom) and from above(top); note that because of the symmetry of the system only a quadrant of the plate isshown. The solid lines show how the undeformed grid (dotted lines) is deformed. Thefinite element simulation shown here demonstrates that this type of buckling cannot bevery large for plates whose thickness to width ratio is more than about 0-1. Thus weconclude that the elongation of the neural plate by active cell interdigitation at thenotoplate and the plate edges plays a subordinate role to the buckling momentsgenerated by the cortical tractor and apical constriction.


shown superimposed. These results are qualitatively similar to those of Odell et al.(1981), who demonstrated that the rolling moment alone is capable of deformingthe neural plate in the proper sequence of shapes: first a flat plate forms, whichthen rolls into a tube.

The simulations in Fig. 11 also show several other interesting features ofneurulation. First, the formation of the plate, and the subsequent rolling of theneural tube, are much more realistically mimicked by the simulations if thenotochord is fixed. This suggests a role for the notochord during neurulation as astructural reinforcement supporting the neural plate. Second, the shapes of theneural tube during rolling are much more realistically reproduced if the rollingcommences at the plate edge and proceeds inward towards the centreline. This,coupled with the hypothesis that basal crawling of plate cells on the epitheliumgenerate the rolling moment, leads us to conclude that the rolling momentcommences at the plate edge, rather than beginning at the centreline, as simulatedby Odell etal. (1981).

Elongation

Simulations have also demonstrated that pure elongation of the lateral bound-ary of the neural plate cannot reproduce the proper shape of the neural fold, andcannot roll the plate into a tube unless the plate is unrealistically thin (cf. Fig. 12and Appendix B). However, in conjunction with the rolling moment, elongationcontributes somewhat to the proper shaping of the neural folds. This is in accordwith the cortical tractor model, which ascribes both the bending moment and theelongation force to the migration of plate boundary cells underneath the adjacentepithelium and the intercalation of cells at the plate boundary.

The sequence of events during neurulation

We have concluded from the simulations that the sequence that best reproducesthe observed shape changes is the following. First, the plate forms by movementsthat occur over the entire neural plate. These movements produce columnariz-ation of the initially cuboidal plate cells, and can be generated by apical con-traction, tractoring, or a combination of the two. Concurrently, elongation of theneural plate occurs first at the centreline and later also at the plate edges. Thiselongation is driven by the active interdigitation of notoplate cells at the centre-line, and the basal crawling and interdigitation of plate cells at the boundary. Thiselongation is not sufficient by itself to generate much of a rolling moment becauseof the thickness of the plate: however, it does appear to contribute somewhatto the rolling forces. The elongation also narrows the plate, which facilitatesneurulation.

Subsequently, rolling of the neural tube commences at the lateral boundaries ofthe plate and proceeds inward (rather than from the centre outward, as suggestedby the simulations of Odell et al. 1981). As we mentioned above, the rollingmoments can be generated by apical contraction or tractoring alone, or incombination. However, the configuration of cells shown in Fig. 10 gives strong


support to the supposition that crawling of the marginal cells on the basalepithelium is a strong component of the rolling moment. This is in accordance withour emphasis on the plate-epidermis boundary as an organizing line for neuraltube formation. We shall report on more extensive numerical studies in a sub-sequent publication.

DISCUSSION

Ettensohn has recently reviewed the various proposals for how epithelial sheetsinvaginate (Ettensohn, 1985). His list of mechanisms includes apical constriction,differential adhesion, cell division and cell rearrangement, amongst others. Wehave proposed here a new mechanism that can drive epithelial morphogenesiswhich we call the cortical tractor model. This model is built around the notion thatthe motile behaviour of cells in an epithelial sheet is similar to that of freelymigrating mesenchymal cells, with the exception that epithelial cells remain firmlyattached at their apical circumference. The term 'cortical tractor' refers to themotion of the cell cortex: cytogel flows in a fountainoid whose source is near thebasal cortex, and whose sink is on or near the apical surface.

The model presumes that membrane and cell adhesive structures are inserted ata basal source, flow apicalward, and are eventually recycled to the cell interior atan apical sink. Junctional structures pile up at the apical periphery becauseresorption cannot keep up with the insertion rate.

According to the cortical tractor model, the motive force for epithelial morpho-genesis derives from the motile activity of the basal and, or, lateral surfaces of thecells. Thus basal crawling generates shear forces on the lateral surfaces betweencells, and constricts the apical surface. This is not exclusive of apical constrictionby circumferential filament bundles; indeed, the tractor motion can provide themechanism that accumulates actomyosin structures in the apical region.

Thus the cortical tractor model shifts the focus of attention from the apicalsurface to the basal and lateral surfaces where motile activity is hidden from directobservation. Indeed, micrographic examination of these surfaces in active epi-thelia does reveal an abundance of protrusive lamellae and filopodia. The time-averaged motions of these protrusions create the fountainoid flow, and provide themotive force driving epithelial deformations by several mechanisms.

First, the tractor motion can generate shear forces between cells whose corticalvelocities differ. These shear forces tend to columnarize the cells, a phenomenonthat usually precedes epithelial folding (Ettensohn, 1985).

Second, if the basal surface can attach to a substratum - for example, anadjacent cell whose tractor speed is slower - then the tractions it develops cangenerate a bending moment. Furthermore, the constriction of the apical surfaceassociated with the tractor motion contributes to this bending moment.

Finally, the cortical motion that sweeps junctional structures apicalward pro-vides a mechanism by which epithelial cells can interdigitate between one another,


changing their neighbours and repacking the sheet without violating the integrityof the apical seal. Active cell rearrangement appears to drive many epithelialdeformations, and the cortical tractor model provides an explanation for how thatprocess can come about.

In the absence of direct in vivo observations, we have attempted to provideevidence in favour of the cortical tractor model via three routes. First, byexamining micrographs of the basal and lateral surfaces of epithelia known to beengaged in morphogenetic activity, we find an abundance of motile appendages.Second, we have examined the mechanical consequences of the cortical tractormotion using mathematical analysis and computer simulations. We find that thepostulated tractor motion can indeed generate the required forces and bendingmoments to produce realistic-looking epithelial foldings. We have used the neuralplate as an example here, where we predicted that cells in the notoplate, or in theneural plate, would move about randomly by tractoring until they interdigitated atthe notoplate-neural plate boundary, where they should remain. Our obser-vations from time-lapse films confirmed this prediction. Finally, the cortical tractormodel is consistent with a wide range of observations on epithelial morphogenesis(cf. Ettensohn, 1985).

Indeed, the cortical tractor model provides the mechanical apparatus necessaryto implement the various other hypotheses of epithelial morphogenesis, includingapical constriction, active repacking and adhesive disparity. For example, theadhesive disparity models of Gustafson & Wolpert (1962, 1967), Mittenthal &Mazo (1983) and others suggest that cell adhesion differences can drive epithelialinvagination. However, adhesive forces cannot by themselves drive cell motions,for adhesion is a force normal to the cell surface. A tangential force is required inorder to bring surfaces with different adhesive strengths into contact. The corticaltractor motion provides the tangential (i.e. shear) forces required to translocatecell surfaces relative to their surroundings. Moreover, the insertion of membraneand adhesive structures at the basal source and their subsequent apical flow andreingestion provides the mechanism for creating and modulating the surfaceadhesion properties of each cell. Thus the cortical tractor model does not replacethe notion of adhesive disparity, but provides the mechanical mechanism forimplementing differential adhesion as a morphogenetic mechanism.

We do not imagine that the cortical tractor model will provide an explanationfor all epithelial morphogenesis. Indeed, we agree with Ettensohn's (1985)conclusion that different mechanisms may predominate in different settings.However, we feel that by drawing attention away from the apical surface, sovisible and compelling, and focusing on the basal and lateral surfaces, whichmanifest so much motile activity, the cortical tractor model can at least provide anew focus for investigating epithelial morphogenesis.

This work was supported by grants NS16072 from NIH to A.G. J., by NSF Grant MCS-8110557to G.F.O. and L.Y.C. and by NSF Grant No. MCS-8301460 to G.M.O. We would like to thank


Raymond Keller for valuable discussions, and James D. Murray who collaborated with us inderiving the placode equations.

APPENDICES

(A) The cortical tractor model for epithelial columnarization

In this Appendix we sketch the basic equations that govern a cortical tractormodel for epithelial deformations (cf. Cheng etal. 1986). The complete equationsfor the cortical tractor model are quite complex. However, by restricting attentionto the vertical force components we can easily write the equations governing theformation of placodes, regions of columnar cells whose formation always precedesfurther folding or invaginations.

(1) Finite difference equations for normal displacements

The simplest model for the cortical tractor can be derived by considering thefree body diagram in Fig. 13. Here we represent two adjacent cells by trapezoidalelements, whose vertices are denoted by (i-l,i, i+1), and refer to the cell withboundaries /, /+1 as cell /. We shall employ the following notation:

Hi = the height at node i (i.e. the boundary between cell i and cell i—1)Ho = the initial (unstressed) height of each cellA, = Ht/Ho = the stretch ratio at node iWi = the width of cell iWo = the initial (unstressed) width of each cellvt = the velocity of the cortical flow in cell ik = the elastic modulus of each cellG = the passive shear modulus of each cell

^elastic

Fig. 13. A free body diagram of two adjacent cells showing the elastic, shear andtractor forces acting on node i (From Cheng, Murray, Odell & Oster, 1986).


a = the active shear modulus for each cell due to tractor motionjU = viscosity of the cortical cytoplasmThe equations for motion for cell i are derived by writing down force balance

equations on node i for the vertical force components; i.e. by equating the sum ofthe forces to the intertial force, md2Hi/dt2, where m is the mass of a cell. However,for this system the inertial forces are negligible (Odell et al. 1981), so that we neednot consider the acceleration term on node i. Thus the force balance equation fornode i is:

FELASTIC + FsHEAR + FACTIVE + FVISCOUS = 0 (1)

which can be approximated by

dHt \(Ht + Ht+1 \Wt

VISCOUS FORCES PASSIVE ELASTIC FORCES

ACTIVE SHEAR FORCESPASSIVE SHEAR ELASTIC FORCES

where k is an elastic modulus and G a shear modulus. Here forces tending toshorten the cell are counted as negative. Note that, in the elastic force term, theterms in parentheses are the elastic forces per unit width of a cell, multiplied by thehalf-width of the cell. This assigns half the elastic force generated by each cell to anode. The active shear force modulus, a, is per unit height, and so is multiplied bythe cell height, Hr Note also that the shear forces depend on the absolute valueof the tractor velocity differences between cells, since only relative motion isrequired to produce active shears.

Since cytoplasm is incompressible, we shall also impose an incompressibilityconstraint on each cell:

IW (3)lorro V /

Thus we can eliminate Wt from the above equations and, after dividing through byHo, the equations of motion can be written in terms of the strain, A,, only.

These equations may be simulated directly as finite difference equations, and weshall present an extensive study elsewhere (see Cheng et al. 1986). Two examplesof the counterintuitive properties of the cortical tractor will be discussed below.

(2) Field equations

Some insight into the behaviour of the model can be gleaned by converting theabove equations into a continuum model. There is no unique correspondencebetween the above finite difference equations and a set of partial differential


equations. However, we can proceed in the most elementary fashion by thefollowing identifications:

A/ > A[X, t), X[+1 — Xf > O

so that we can expand A in a Taylor's series to second order, e.g.

aX 1

This yields the following nonlinear partial differential equation for X(x, t):

(4)

where we have employed the constant volume constraint and defined the followingquantities:

ao = ad

The linearized version of equation (4) is easier to understand and captures theessential features. Therefore we can substitute A = 1+w into the above equationand retain only linear terms in u, the height perturbation. Thus for small velocityperturbations, v(x), the linearized cortical tractor model is:

du , d u dv d vdt ° dx^ ° dx dx2

where D = G1- K\.The first term on the right-hand side of this equation is the passive elastic force

tending to restore the cells to their unstrained cuboidal shape. The second term isthe elastic forces tending to smooth out variations in cell height. If G > kWo/2 thisterm is always stabilizing; however, if G<kWo/2 the system may becomenumerically unstable under perturbations whose wavelength is shorter than thewidth, Wo, of the finite element. This is simply an artifact of the continuum limit:physically, the passive elastic forces cannot generate deformations. The third andfourth terms arise from the active shearing stresses generated by the corticaltractor. The third term is an active 'convection-like' force which tends to deformthe sheet wherever there is a gradient in tractor velocity, v. The fourth term is theactive shearing force which tends to amplify height variations.

Note that because of the incompressibility constraint, the taller a cell grows,the narrower it becomes. Therefore, the strain A is proportional to the celldensity. Therefore, we can substitute for u the normalized cell density deviation


(N—No)/No, where No is the undeformed cell density (i.e. when A = 1, u = 0), andobtain an equation in cell density, N, which has the same form as equation (5).

If a cortical tractor velocity, v, is prescribed by some means (e.g. a boundarycondition, or a chemical gradient) then equation (4) predicts a variation in celldensity, and therefore in cell shape, wherever variations in v arise. This is becausethe active shear forces which tend to deform the cells can only arise when gradientsin v are present. The consequences of this simple mechanical effect are quitesubtle, as discussed in detail by Odell & Bonner (1986) where a similar model wasapplied to the morphogenesis of slime moulds. In the present setting, it is worthnoticing that a constant linear gradient in the tractor velocity, v, produces auniform placode («>0) , as shown in Fig. 14A. Furthermore, an asymmetricvelocity profile (e.g. uccsin(x)) produces an asymmetric deformation which isdisplaced, as shown in Fig. 14B. Since local chemical conditions regulate the

Fig. 14. The deformation fields corresponding to two simple tractor velocity fields.The tractor velocity is a monotone function of local chemical conditions, and so thevariation in v(x) can be taken as reflecting the spatial chemical concentration distri-bution. (A) A uniform gradient in tractor velocity, v, produces a placode of constantheight: i.e. a constant displacement field u(x), wherex measures distance along the cellsheet and u measures the height perturbation from the uniform, unperturbed state.The taper at the edges is due to the imposition of zero displacement boundary con-ditions. (B) A periodic tractor velocity field, v(x), produces a displacement field, u(x)which is shifted with respect to the velocity field (from Cheng, Murray, Odell & Oster,1986).


tractor velocity, v, this implies that the cortical tractor 'reads out' chemicalconditions in a somewhat counterintuitive fashion.

(B) The finite element shell model of the neural plate

The finite element method

The simulations of neural folding shown in Figs 11 and 12 were performed usinga finite element model for shell-like bodies (Zienkiewicz, 1977). This method isbased on a variational formulation, which minimizes a functional representing thetotal energy of the mechanical system. A material body is first subdivided into acollection of sub-bodies, each of which is modelled by a finite element thatrepresents its geometric and mechanical characteristics. One then computes thecontributions to the energy functional from all of the elements, and then mini-mizes this functional to obtain the geometric configuration of the complete body.No restrictions need be placed on the shapes and mechanical properties of theelements, and so systems with complicated geometry and physical properties maybe analysed.

In order to formulate the energy functional which characterizes the system, onemust account for three kinds of relationships:

(1) The equations of mechanical equilibrium. These relate the external forcesapplied to the system to the pattern of internal stresses.

(2) The kinematic relationships, which relate the strains within the material tothe displacement of material points of the body.

(3) The constitutive relation which specifies the mechanical properties of thematerial by a stress-strain equation.

In applying the finite element method to the neural plate, we have specialized tothe case of thin-shelled bodies. This is because the thickness of the neural plate israther small compared to the plate radius (i.e. a thickness/radius ratio of less than10 is sufficient).

The neural plate model

We have studied several possible mechanisms of neural plate deformationseparately in order to assess their relative contributions to the observed morpho-logical changes. The combined effects of apical constriction (via apical filamentbundle contraction and, or, tractoring) and basal crawling is to produce a rollingmoment. Elongation of the notoplate can also induce a rolling moment bygenerating a transverse compression (Toisson buckling'). First, we study the purerolling problem employing the axisymmetric shell elements developed by Cheng(1986). The curved plate geometry was achieved by modelling the embryo as atorus, with a very large major radius. The thickness/radius ratio was taken as 1/10for the epidermal cell layer, and twice that for the neural plate. These representthe approximate size proportions at the time of tube formation. Although theeffects of apical constriction and basal crawling both create an active rollingmoment, they were modelled in different ways. The effect of apical constriction


was imposed over the entire plate simultaneously, whereas basal crawling initiatednear the plate margin (cf. Fig. 10), and progressed toward the midline, since thispattern of moments produced the most realistic sequence of plate geometries.

In order to investigate the role of midline and, or, marginal elongation a fullthree-dimensional model is required. We employed for this a general purposefinite element program called ABAQUS (Hibbit, Karlsson & Sorensen, 1982).The neural plate was modelled as flat plate, as shown in Fig. 12.

A tangential force was applied to the plate along the midline to produce theelongation. A small perturbation in the vertical direction (modelled as a uniformlydistributed vertical pressure) was introduced to initiate the buckling instability.Plate thicknesses around 1/20 and 1/200 were employed, and the simulationsshowed that Poisson buckling was only significant for quite thin plates.

In the above analyses, the elastic properties of the epithelial cells was rep-resented by a Hookian (linear) constitutive relation, and viscous effects wereignored. In all cases the formulations we used were valid for large displacements.A complete description of the simulations will be published elsewhere.

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(Accepted 1 April 1986)

Neurulation and the cortical tractor model for epithelial folding

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